The tangents to the parabola $y^2 = 4ax$ make angles $\theta_1$ and $\theta_2$ with the positive $x$-axis. If $\cot \theta_1 + \cot \theta_2 = c$,then the locus of their point of intersection is

If a double ordinate of the parabola $y^2 = 4ax$ has a length of $8a$,then the angle between the lines joining the vertex of the parabola to the ends of this double ordinate is ............... $^\circ$.

The axis of a parabola is along the line $y=x$. The distance of its vertex $A$ from $(0,0)$ is $\sqrt{2}$ and that of its focus $S$ from $(0,0)$ is $2\sqrt{2}$. If $A$ and $S$ lie in the first quadrant,then the equation of the parabola in parametric form is

$A$ tangent to the parabola $y^2 = 8x$ makes an angle of $45^\circ$ with the straight line $y = 3x + 5$. Find the equation of the tangent.

The focal distance of a point $P$ on the parabola $y^{2}=12x$ if the ordinate of $P$ is $6$,is

Let $L_1, L_2$ be the lines passing through the point $P(0,1)$ and touching the parabola $9x^2+12x+18y-14=0$. Let $Q$ and $R$ be the points on the lines $L_1$ and $L_2$ such that the $\triangle PQR$ is an isosceles triangle with base $QR$. If the slopes of the lines $QR$ are $m_1$ and $m_2$,then $16(m_1^2+m_2^2)$ is equal to ..............